Model Answers to the Third Homework
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چکیده
1. Chapter 4, §3: 5. I and A are additive subgroups, and the intersection of subgroups is a subgroup, so that I ∩A is an additive subgroup of R, whence of A. Now suppose that a ∈ A and i ∈ I ∩ A. Then a ∈ R and I is an ideal of R, so that ai ∈ I. On the other hand, A is a subring of R, so that ai ∈ A as i and a are in A. Thus ai ∈ I ∩ A. It follows that I ∩ A is an ideal. 6. As in the previous question, I ∩ J is an additive subgroup, as both I and J are. Suppose that r ∈ R and a ∈ I ∩ J . As a ∈ I and I is an ideal, ra ∈ I. Similarly ra ∈ J . But then ra ∈ I ∩ J and I ∩ J is an ideal. 15. Suppose that a ∈ R. Then a ∈ IJ if and only if a has the form i1j1 + i2j2 + · · ·+ ikjk, where i1, i2, . . . , ik and j1, j2, . . . , jk are in I and J respectively. It is therefore clear that IJ is closed under addition and it is clear that IJ is non-empty. Thus IJ is an additive subgroup. Suppose that r ∈ R and a ∈ I. Then
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